John said:
Yes, but it's not the whole story.
It is not my whole post, either, since you cut the quote off early.
I know it wasn't your whole post let alone your whole argument but it
was enough for me to hang my toy example from.
I suspect you double-counted the 9 possibilities where each person
gets 1 item, and also the 6 possibilities where 1 person gets two
different items. 36 - 9 - 6 = 21.
My reasoning in more detail was that one dollar can be spent in six
ways:
(Alice, burger); (Alice, fries); (Alice, shake); (Bob, burger); (Bob,
fries); (Bob, shake)
The spending of the two dollars is independent so the total number of
ways they can be spent is 6x6 = 36.
However, I think that you're right as burgers are indistinguishable
from each other, as are portions of fries, as are shakes, at least in
a simple toy model. I was counting the case in which the first dollar
buys Alice a burger and so does the second as two cases rather than
one. As you said, there are six such cases that I've counted twice. I
was also counting cases in which the first dollar buys Alice a burger
and the second buys Alice fries as distinguishable from the one in
which the first buys her fries and the second a burger. If they're
indistinguishable it's clearer to describe them as "Alice doesn't have
a shake" or whatever and there are actually only 3x3=9 cases rather
than the eighteen that I counted. So the correct count is 36-6-9 = 21,
as you calculated.
Your method of counting has the virtue of being more elegant as well
as the greater virtue of being correct. Thanks for the correction.
Also, if each person chooses one of 7 uniformly, the 28 outcomes will
not be uniform: for example, Bob with 2 burgers will be half as likely
as each with a burger. It seems that the outcome will be less
predictable, more randomized.
Yes, that's true. There will be some quite odd cases in which Alice
buys Bob a burger and vice versa too (and similarly for the other two
products).
Do you think Nick would argue the same thing (Alice must give everyone
a dollar) if Alice had $10 and 9 others had no dollars? What if Alice
had $20 and ten others had $2 each? What if, instead of dollars, we
had coupons for a medical treatment to extend life by a year? Must
Alice give up years of her life? What about contracts to provide 1
year of manual labor to XYZ corporation? If Alice was liable for 2 of
those contracts, and Bob was liable for none, must Bob take 1 of the
contracts? What would you guess Nick would argue?
I think that in the cases with the money or the coupons Nick would
argue that Alice should be made to give to the others, but not in the
case with labour contracts, but I suppose we'll have to wait for him
to give his opinion. Of course, not all years of extended life have
the same cost in expended resources so that example's a bit strange.
Similarly, the opportunity cost of making different people engage in
manual labour varies wildly.
Rich
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